Twisted cotangent symplectic form

If $\sigma$ is a closed two-form on $X$, then $\omega +{\pi }^{\ast }\sigma$ is a symplectic form on $T^{\ast }X$, called a twisted cotangent symplectic form. Nondegeneracy follows from the horizontal-vertical block form of the canonical symplectic form.

If $\sigma =d\beta$, cotangent fiber translation by $+\beta$ gives a symplectomorphism from $(T^{\ast }X,-d\alpha )$ to $(T^{\ast }X,-d\alpha +{\pi }^{\ast }\sigma )$. Its pullback changes $\alpha$ by ${\pi }^{\ast }\beta$, so the extra two-form cancels with $-{\pi }^{\ast }d\beta$. The opposite translation gives the inverse direction, globally without compactness.

With $\omega =-d\alpha$ and ${\omega }_{\sigma }=\omega +{\pi }^{\ast }\sigma$, the one-form graph of $\theta$ is a Lagrangian submanifold exactly when $d\theta =\sigma$. Indeed its pulled-back symplectic form is $\sigma -d\theta$ and its dimension is half that of the total space. This extends graph of a closed one-form is Lagrangian.

If $[\sigma ]\ne 0$ in de Rham cohomology, a twisted cotangent symplectic form has no Lagrangian submanifold whose projection is a diffeomorphism onto the entire base. Such a submanifold would be a global one-form graph, and the twisted Lagrangian graph criterion would make $\sigma$ exact.